Phyllotaxis
the spirals are counted, not admired
Concept

Null model — where it appears

A pattern generated with no rule in it, against which an ordered arrangement's statistics are read. Several observables here turn out to be reproducible by one, which is the strongest way of showing that the observable was not evidence for the rule.

Named by 42 essays across 7 fields — each of them below, with the objects they name alongside it.

The exponent fitted from the junctions, rather than assumed. Sweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.

Fitting the exponent

Assuming the exponent is three and reporting the error says how far the data is from that assumption. Fitting the exponent and reporting what it comes out as says what the network is doing — and an estimator has to be shown returning something other than three, or it is not a fit.

branching · Exponent The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

The second moment is the measurement

The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. On heads of nine hundred organs the mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

tissue · Sixsides A lattice with independent errors. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, independent errors — that earlier work's control at 0.5° of independent scatter. The largest comb mean is 0.029 against a sampling band of 0.073, and the readout refuses.

A disturbance with a memory

That earlier work's control assumed that a plant's errors are independent from organ to organ, and nobody had tested it. Give the errors a memory — each one a fraction of the last, up to a coefficient of 0.97 — and the comb does not appear. The obvious threat to the result turns out to be empty, and the algebra says why before the measurement does.

mechanism · Noise colour A lattice with an error inherited from the two contact neighbours. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, an error inherited from the two contact neighbours at coupling 0.7. The largest comb mean is 0.514 against a sampling band of 0.073, and the readout returns 8/13.

Errors that pass between organs

An organ's neighbours are the ones eight and thirteen places back — that is what a parastichy pair is. So a disturbance transmitted by contact is correlated at exactly the two lags the readout examines, and it does not have to be told them. Driven into a lattice with no rule in it, it returns the counted pair on eight stems out of eight.

mechanism · Noise transport The two combs, in the proportions the rule gives them. The ratio of the second comb to the main one, for a kinematic lattice whose errors are inherited from its two contact neighbours, against how unevenly that inheritance is split. The horizontal line is where the placement rule's own stems sit, at 0.65. Weighted by distance — the coupling a d⁻³ interaction would give, which at this rise favours the 13-neighbour by 1.26 to one because the 13-hop is the shorter — the forgery sits at 1.46, well above the rule. It reaches the rule's value only at about 3 to one the other way, which is a factor of 4 against what distance supplies and in the opposite direction.

What a forgery has to know

A lattice with transported errors reproduces the comb and the pair, so one quantity is left: the two combs' relative strength. Weighted by distance the forgery puts more in the second comb than the first; the rule does the opposite. It matches only if the coupling is turned three to one towards the further neighbour, which no falloff supplies.

mechanism · Noise transport Murray's law and Da Vinci's become one measurement at 12% of radius error on the informative band. Two trees measured the same way: 50 junctions of daughter ratio 0.6–1, 300 replicate samples at each of 15 error levels, one tree built at exactly 3 and one at exactly 2. Each shaded band is the central 90% of the recovered exponents. Both run downward, but the tree at 3 runs down faster — -113σ² against -29σ² — so the two close on each other. At 11% they are still apart; at 12% the bands overlap and one study's answer could have come from either tree; at 20.5% the means cross, and above it a tree built at 3 measures lower than a tree built at 2.

Where three and two become one

Two trees, one built to obey Murray's law and one to obey Da Vinci's, are measured through the same fifty junctions with the same instrument. At twelve per cent of error on each radius the two answers overlap, and above twenty and a half the tree built at three measures lower than the tree built at two.

branching · Exponent error A lattice with an error that repeats every 8 organs. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, an error that repeats every 8 organs at period 8, weight 0.7. The largest comb mean is 0.592 against a sampling band of 0.073, and the readout returns 8/10.

A periodicity is not a lattice

Give a lattice's errors a period of eight and a comb appears at spacing eight, on an arrangement with no rule in it. But the partner it names is 10, then 12, then 11, then nothing — an accident of the disturbance rather than a measurement of the pattern. The forgery is caught by reading a second stem, and by nothing else.

lattices · Noise colour Take away the organ eight places back, and the next one goes into the hole. The last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.

The organ that was taken away

Every observable this site has is read off an arrangement that was finished before the reading began, and earlier work here showed what that costs. So remove one primordium from a settled stem and place the next one against what is left. The rule has to answer. The rival account cannot, because in it no organ's position was ever computed from its neighbours.

mechanism · Ablation The ratio follows the disturbance, not the rule. The ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.

The ratio was never about the rule

The comb has already been retracted here as evidence that a plant computes its pattern, and one quantity was exempted from the retraction: the ratio of the two combs, which a placement rule and a transported disturbance divide differently. Drive seven disturbances through the same rule and the ratio spans 0.45 to 1.09. The exemption does not hold, and the angle sequence has nothing left.

mechanism · Noise transport The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

A dip with no outer edge

The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.

tissue · Second statistic Two combs, at a rise of 0.005. The autocorrelation of 760 divergence angles from one stem held at a rise of 0.005. The filled teeth are the lags at multiples of 8; the open teeth are the second comb, at the same spacing offset by 5. Reading the spacing off the first and the offset off the second gives the pair 8 and 13, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.

A disturbance that is not passed on

The disturbance that forges every observable measured here does two things at once — it correlates an organ's error with its contact neighbours', and it hands that error on to be handed on again. Every result about it has been unable to say which half did the work. This is the control that takes the second half away and keeps the first.

mechanism · Noise colour The damage is the sharing; the forgery is the history. Three disturbances of the same size, measured four ways. The two left columns are stems grown by the placement rule and jostled at 0.25° per organ: a disturbance shared between the contact neighbours scatters the lattice by 0.71° against white noise's 0.57°, and one inherited from them — the same sharing, passed on again at every organ — by 0.97°. The two right columns are kinematic lattices with no rule in them at all, where the whole question is what a disturbance can manufacture. The inherited one returns the pair on 8 seeds of 8 with a main comb of 0.205 against a band of 0.073; the shared one, at the same coupling and the same scatter, returns it on 1 and makes a comb of 0.099, which is the band. So sharing an error with the organs you touch does the damage, and only passing it on and on forges the evidence.

The forgery needs a history

A disturbance passed between touching organs manufactures the comb, the second comb and the parastichy pair on an arrangement with no rule in it — which is why the comb stopped being evidence. Give the organs the same correlation with no accumulation in it and the forgery collapses: one seed in eight returns a pair, and the comb is the noise floor.

mechanism · Noise transport The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 55, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0067° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 595 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.

The window is the neighbour

An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.

tissue · Second statistic The wander is in the disturbance and not in what a plant lets you measure. Each disturbance measured twice, in the same statistic. On the left, the variance of the block means of the disturbance's own deviates, over blocks of 100, as a multiple of what independent draws would give; on the right, the same quantity for the divergence sequence those deviates produce, over blocks of 128. The left column is what this site measured when it proposed a slow wander as a second observable. The right column is what a botanist would have: a divergence is the difference of two organs' errors, and differencing is exactly the operation that removes power at low frequencies. The disturbance inherited between touching organs goes from ×49.1 — the largest here — to 0.83, which is what independent errors give. The one with a memory in time keeps most of its own.

A difference forgets a drift

This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift.

mechanism · Noise transport Seven fractions with one neighbour distance and every denominator. Each member of a matched set drawn on its own stretch of the divergence axis, 0.5° either side of itself, with the nearest other rational marked. The distances are 0.3158°, 0.3158°, 0.3117°, 0.3069°, 0.3117°, 0.3077°, 0.3064° — a spread of 3.1% — while the denominators run 19, 20, 21, 23, 33, 45, 47, a factor of 2.47. That is the construction this thread needed. Every instrument for the width of a disorder dip has a free parameter set by how close the neighbour is, so a hypothesis about the neighbourhood cannot be tested by varying the neighbourhood; on this set the neighbourhood is held fixed and the arithmetic of the fraction is what varies.

Fractions with the same neighbours

Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them.

tissue · Second statistic Through the rule, the drift survives and the inheritance still does not. How much of a divergence sequence's variance survives being averaged over blocks, on stems the rule grew. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a differenced stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 46 at a block of 64. The ones inherited between touching organs do not climb at all — 1.51 and 1.90 at the same block — although their own deviates carry ×— and ×— an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.06 and 0.12.

What the rule does to a drift

A placement rule was supposed to leave no slow wander in a divergence sequence, because its errors are corrections rather than inheritances. Driven by a disturbance that drifts, it leaves a larger one than a lattice with no rule in it at all — while cutting the per-organ scatter by more than half. The rule removes what is relative between neighbours, and a drift is not.

mechanism · Noise colour 264 exceptions in a 2400-organ head, and eight circles. A golden, 137.508° head of 2400 organs, tessellated inside 86% of its radius. Every interior cell is drawn, and the 135 five-sided cells and 129 seven-sided ones are marked apart from the 1367 hexagons, and the pale circles are radii computed from the divergence angle through the lattice's third-shortest vector — nothing is fitted. Every defect sits within 0.64 of a cell spacing of one of those eight circles, and between them there is not one exception in hundreds of cells.

The defects lie on rings

The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.

tissue · Topological charge One tree sized for flow and for stress, with each branch 2^(−1/2) the length of its parent. The same symmetric tree, 8 generations deep, each generation's branches 2^(−1/2) the length of the one before and turned 30° at every fork, sized two ways and drawn to one trunk width. On the left each branch's radius cubed is proportional to the tips it feeds — Murray's flow rule — and every junction conserves r³. On the right each branch is sized so that the same load on every tip bends it to the same stress at its base, radius cubed proportional to the sum of its lever arms to its tips; its trunk junction conserves r to the power 1.967, its outermost junctions 1.349, against a deep-tree limit of 2.000. The two trees thin at different rates from the same trunk.

A cube law with a lever arm

Murray's exponent of three comes from moving fluid for the least work, and Da Vinci's two has had no derivation here, only the name of the mechanical answer. Size every branch so that the same wind on every tip bends it to the same stress, and a junction conserves r to the power 3/(1 + log₂(1/λ)), where λ is how much shorter each branch is than its parent. A crown that fills a plane gives exactly two; halving lengths gives one and a half; no shortening gives three. Murray's flow rule gives three at every λ, so the lengths of a tree's branches say which mechanism sized it.

branching · Murray A periodicity reports a different partner every time. eight kinematic lattices, differing only in the seed of their disturbance, each read by the same instrument. The disturbance repeats every 8 organs at a weight of 0.9: it puts a strong comb at spacing 8 — 0.75 against a band of 0.07 — and the partner it names is 8/10, 8/12, 8/11 across the 8 stems and never 8/13, which is what the position counter finds in every one of them. There is no placement rule in any of these arrangements.

The control a survey would need

A comb no longer shows that a plant computes its pattern, so the survey this site has been specifying for a long time has to change. What it loses is its headline; what it gains is a measurement a botanist can actually make — six requirements, four of them already in the specification, and a quantity nobody has ever reported.

wrong · Control spec A deeper rule passes more of a drift, not less. The wander left in a stem's divergences, against how many organs its disturbance stays correlated over, for rules whose neighbourhoods run from 3 organs to 182. The prediction under test said a rule should pass a drift once the drift outlasts its neighbourhood, so the shallow rules should be the leaky ones and each line should turn where its own depth is crossed. Every line rises smoothly and the deepest rule is the highest of them at every correlation length — 82 against 22 at the longest drift. There is no crossover anywhere in the sweep.

The drift goes the other way

A rule that corrects what its neighbourhood shares should let through any disturbance slower than its own reach, and should suppress anything faster — a crossover, tracking the depth. Swept over a neighbourhood that changes by a factor of sixty, there is no crossover anywhere, and the deep rule passes nearly four times as much as the shallow one. The prediction is not weakly supported; it is backwards.

mechanism · Noise colour Every five is a cell away from a seven, and the loneliest is 0.927 spacings out. How far a five-sided cell is from the nearest seven-sided one, in cell spacings, on a golden, 137.508° head of 2400 organs. The measured bar runs from the closest five to the loneliest — 0.833 to 0.927, with a median of 0.919. It stops a single cell out, so there is no unpaired tail at all rather than a small one. The nulls are seeded permutations over 200 draws: relabelling which defects are fives puts the average five 1.254 ± 0.064 spacings away, and scattering the whole multiset over the interior cells puts it 1.786 ± 0.092. 100.0% of the fives share a wall with a seven against 68.2% for the strong null, z = 7.3.

Every five is bound to a seven

A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.

tissue · Topological charge The next organ moves for the last 13, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.

An experiment a needle could run

For eight instalments the outstanding item has been a survey — photographs, a protractor, hundreds of specimens — and it has not been done. The intervention is a different kind of ask, and a cheaper one: a needle, one apex, and a yes-or-no per ablation. Here is what it would cost, what it would settle, and the four ways it could come out.

wrong · Ablation Which arrangements carry a comb, and what each one reports. The largest comb mean in five arrangements at a rise of 0.005, all read by the same instrument at the same length, with the sampling band of 0.073 marked. Only the first is a placement rule; the other four are kinematic lattices with no rule in them, differing from one another only in how their azimuth errors are structured. Independent errors and errors with a memory leave nothing to read. A repeating error puts up a comb and names a partner that is not the lattice's. Errors inherited from the contact neighbours reproduce both the comb and the pair.

The comb was never the rule

A control is only as strong as the alternative it builds, and the earlier work built one that varied the rule while holding the disturbance fixed at independence. Five rounds of the angle-sequence thread, with what each claimed and what still stands — and why the next evidence has to come from an intervention rather than from a longer stem.

cylinder · Noise transport The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 9 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.28, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.

The survey loses its second outcome

The survey specification written earlier here names three results the survey could return, and the second — a ratio near or above 1.30, read as evidence against the placement rule — is the one that would have been worth publishing. It does not survive the measurements here. The ratio moves with where the plant sits between two transitions, and it moves again with the colour of the plant's own disturbance.

wrong · Survey spec

The order belonged to the method

A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.

wrong · Second statistic

A crown that carries its own wood

Sizing every branch so that equal loads on the tips bend it to one stress gives a crown filling a plane Da Vinci's exponent of two. Move the load onto the wood and the sizing becomes a fixed point, because a branch's load now depends on the radii being solved for. Under the wind on its wood a planar crown still conserves two, but only as a limit its trunk is two tenths short of at fifteen generations. Under its own weight it conserves one — radius rather than area, the stress-similarity law that radius goes as length squared — and a crown carrying leaves and wood reads the leaves' two near its twigs and the wood's one at its trunk, with the handover set by how much of the trunk's load the wood carries.

branching · Murray

The ablation a plant would survive

The intervention proposed earlier returns a spiral count from a yes-or-no answer, needs no protractor, and was specified at one rise. Measured across the ladder it acquires three conditions a real experiment would have to meet — and one of them is that the plant must not be too coarsely patterned, or nothing will go wrong at all.

wrong · Survey spec

The ratio was the floor of a curve

One number was left standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest.

cylinder · Noise transport

Matching instead of correcting

Two rounds of work failed on one question because every instrument's free parameter was set by the thing under test. The repair was not a better instrument or a model of the bias: it was choosing what to compare so that the confound could not vary. That move is available in four other places here, and three of them have already used it without anybody naming it.

wrong · Instrument ceiling

A second moment that goes to zero

The mean squared departure of a cell's side count from six separates a random tissue from a whorled head by a factor of eighty, on heads of 900 organs. Read at thirty-three head sizes it is exactly the share of cells on the defect rings of a spiral head and falls as one over the square root of the organ count, it falls as one over the count on a whorled head, and the factor is 25 at 300 organs and 677 at 8,000.

tissue · Sixsides

A cut of two organs

One organ removed from a stem is felt out to the larger parastichy number and no further, and at the coarsest arrangement the stem always repairs itself — so the one rung where the interesting prediction could be checked had no experiment that could reach it. Two organs can. The second cut brings a parameter with it, and that parameter turns out to be a control.

emergence · Ablation

One way round, seventeen times

The two chains that change places in a wrecked stem are adjacent, which is symmetric and says nothing about direction. Label them by lag from the hole and the one displaced forwards is always the lower of the two — on every row of the census, without an exception.

mechanism · Damage shape

Four accounts of one angle

The exchanged pair in a wrecked stem is about one divergence step, and about is doing twelve per cent of work. Four candidate units were written down and scored on the same seventeen rows: the cut stem's own step, the surviving family's step, the control's step, and the control's corrected.

mechanism · Damage shape

The recount aims where the counter expects

A counter who recounts an announced reading knows it went wrong, and if the same habit spoils both counts of a head, the first error says where to aim the second. But the reading alone does not say which way the first erred: a reading of 34/54 is as well explained by a whorled 34/54 read right, or by 34/53 read long, as by 34/55 read short. The direction comes from what the counter expects. Expecting Fibonacci, an aimed recount at 7.2° and a habit correlated at 0.9 reads 34/55 69.6 per cent of the time where an unaimed one reads it 41.8, and the census needs seven specimens rather than fifteen. Expecting only a spiral, it aims the wrong way and reads 34/55 5.5 per cent of the time. The belief that helps is the hypothesis the census is testing: uncapped, it reads the geometry's whorled 3/6 heads as 3/5 and a census of a hundred rejects a true null 40 per cent of the time; capped, it still reads a silent 33/53 as Fibonacci twice as often. And no aimed recount spends fewer counts than 13/21 counted once.

wrong · Sample size

Three organs and no mirror

A coarse stem cut of two organs can end up as its own mirror image — the same lattice wound the other way, counts unchanged, handedness reversed. Finer stems never do it, and two accounts of why were on the table: coarseness, or the share of the neighbourhood removed. A three-organ cut at the finer arrangements settles it, and the answer is the first.

emergence · Dose

The share was not the thing

Two organs out of a front of five reverses a stem's handedness; three out of eight does not, and neither does five out of eight, which is a larger share of a larger neighbourhood. The hypothesis under test was that the dose decides the destination. It decides whether a stem falls off its lattice and nothing about where it lands.

emergence · Dose

The alternation is not a period

Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing.

cylinder · Rung interior

Six of six is not a measurement

A panel comparing two rules seed by seed reported the deeper one winning every one of six seeds at four correlation lengths out of six, and was read as a lattice with no corner in it. Six of six is the largest number the panel can print, so the flat middle was a reading of the ceiling — and raising the disturbance brings a corner out of it.

emergence · Noise colour

A steeper rule walls nowhere else

The account of the wall at the fine end was that the basin narrows because the neighbourhood deepens, which predicts a steeper falloff walling somewhere else. Grown at four exponents, the four columns settle 30, 31, 29 and 27 of 72 — a spread of 0.056 against an error of 0.058.

emergence · Settling

A wall that stopped moving

Four falloff exponents were reported not to move the rise below which stems stop reaching a lattice. Their measured walls spanned a factor of two and ordered themselves 2, 5, 3, 4. At twenty starting angles they span a fifth of one and order themselves 5, 4, 2, 3 — so the conclusion was right and its arithmetic was noise.

emergence · Exponent

Round numbers are not a sample

The nine starting angles the settling table was grown from reach a lattice four times in ten. Eight angles placed exactly halfway between them reach one a quarter of the time. The difference is not noise and it is not the range — several of the nine sit next door to somewhere a stem could settle.

emergence · Settling

Twenty angles instead of nine

Every claim in this collection about where a stem ends up rests on nine starting angles a cell, and the file that uses them says so — it computes a binomial error of 0.17 and declines to read a spread against it. Eleven more angles halve that error and change what several of the numbers were.

emergence · Settling

Named alongside it

The objects these essays reach for when they reach for this one.

Honest limitsMeasurementDiscriminationArtefactEvidenceFalsifiabilityAutocorrelationParastichy pairMeasurement errorClaim testingThe placement ruleNoise

All concepts